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336 XI ELEMENTARY SPECTRAL THEORY

(b) Suppose in addition Uis continuous (cf. Problems 3(c) and 4). Deduce from (a)

that \\U|| = sup (Ox | x).

11*11^1

7. Let F, G be two separable complex Hilbert spaces, (a_n) (resp. (b_n)) (n ^ 1) a Hilbert
basis of F (resp. G), L the Hilbert sum (Section 6.4) of F and G. Let v be the con-
tinuous operator in L defined by v(a_n) ~ 0, v(b_n) = a_n/n, and let E = v(G) -f v*(v(G)).
Let u be the restriction of v to E. Show that u is compact and has an adjoint, but
that w* is not compact. (Observe that v(Q) is dense in F but not closed in F; if (x_n) is a
bounded sequence of points of u(G) converging to a point in F, but not in v(G), show
that the sequence (u*(x_n)) converges to a point in L which is not in E, using the fact
that the restriction of v* to F is injective.)

8- The notations and assumptions are those of (11.5.7). Let (A_n) be the decreasing
sequence of numbers >0 such that, for each k, the number of indices n such that
A_n = jLt_fc is equal to dim(E(/z*)); let (a_n) be an orthonormal system in E such that, for
the indices n for which X_n = /x_fc, the a_n constitute a basis of E(jn_fc). We say that (A_n) is the
full sequence of strictly positive eigenvalues of u.

(a) Show that A_n is the maximum value of (u(x) \ x) when x varies in the subset of E
defined by the relations ||*|| = 1, (x \ a_k) = 0 for 1 < k ^ n — 1; furthermore, that
maximum value is attained for x = a_n (use (11.5.7(d))).

(b) Let z_i9..., z_n_i be arbitrary vectors in E, and denote by P_M(ZI, ..., ^«_i) the
l.u.b. of (u(x)\x) when x varies in the subset of E defined by the relations HJC|| = 1,
(*|z_fc) = 0for 1 ^k^n-1. Show that X_n = p£ai₉...₉a_n-i)^p₁ti₁,..._tz_n-₁) (the
" maximinimal principle "; take x in the subspace generated by a^,..., a_n and verifying
the relations (x \ z_k) — 0 for 1 =£ k ^ n — 1).

(c) Let «', u" be two compact self-adjoint operators, and suppose « = «'-f-«";
let (AO, (A£) be the full sequences of strictly positive eigenvalues of u' and u'\ respec-
tively, (a'_n) and (a'n) the corresponding orthonormal systems. Show that if A_p, A£ and
A_p+€_! are defined, then X_p+q^i < A£ 4- A£ (consider p_H(ai,..., «J_i, ai,..., a'q-i).
If the sequence (Xn) is finite and has N terms, and if Ap and A_P+N are defined, then
X_p+_N =s$ A; (same method, observing that (u"(x) \ x) ^ 0 if (x \ tf) = 0 for 1 ^ j ^ N).

(d) Under the same assumptions as in (c)_s show that if A_p and X_p are defined, then
|A_P — A;^| ||«*|| (use the relation X_p = p_u(a_lt ...,a_p^i)). Furthermore, if u"2*Q
(resp. u'' ^ 0), then A_p ^ A_p (resp. A_p < A_p) (same method).

(e) When E is finite dimensional, transcribe the results of (b), (c), (d) for hermitian
forms on E x E (see Problem 3). Apply to the following problem: let /, (1 ^ i ^ n)
be regulated functions in a compact interval I = [a, b], and let I' = [c, d] be an

interval contained in I; let A = detQ*fifjdt}, A' = det/ f" f,fj dt\ be the Gram

determinants corresponding to I and I⁷; show that A/ < A, by expressing the Gram
determinants as products of eigenvalues.

9. (a) Let M be a compact self-adjoint operator in a complex Hilbert space E. Let
H be a closed subspace of E, and p the orthogonal projection of E onto H (Section
6.3). Show that the restriction v to H of p o u (or ofp o u o _p) is compact and self-adjoint
and that (v(y) \ y) = (u(y) \ y) for y e H (use the relation p* = p). Let (A_M), (^_n) be the full
sequences of strictly positive eigenvalues of u and v respectively. Show that if A_nand fji_n are defined, then fi_M ^ X_n (use Problem 8(b)).

(b) Suppose in addition u is positive. Show that for any finite sequence (x_k)^_k^_nof points of E, det((«(*i) | *,)) < AiA₂ • • • A_n det((x_t \ Xj)) (apply (a) to the subspace H
generated by x_lt..., x_n).

10. (a) Let u be a hermitian operator in a complex prehilbert space E. Show that
for any integer «>0, and any xeE, \\uⁿ(x)\\² < \\if~^l(x)\\ - ||M^W+I(JC)|| (use Cauchy-
Schwarz).



	336 XI ELEMENTARY SPECTRAL THEORY (b) Suppose in addition Uis continuous (cf. Problems 3(c) and 4). Deduce from (a) that \\U\|\| = sup (Ox \| x). 1111^1 7. Let F, G be two separable complex Hilbert spaces, (a_n)* (resp. (b_n)) (n ^ 1) a Hilbert basis of F (resp. G), L the Hilbert sum (Section 6.4) of F and G. Let v be the con- tinuous operator in L defined by v(a_n) ~ 0, v(b_n) = a_n/n, and let E = v(G) -f v(v(G)).* Let u be the restriction of v to E. Show that u is compact and has an adjoint, but that w* is not compact. (Observe that v(Q) is dense in F but not closed in F; if (x_n) is a bounded sequence of points of u(G) converging to a point in F, but not in v(G), show that the sequence (u(x_n))* converges to a point in L which is not in E, using the fact that the restriction of v* to F is injective.) 8- The notations and assumptions are those of (11.5.7). Let (A_n) be the decreasing sequence of numbers >0 such that, for each k, the number of indices n such that A_n = jLt_fc is equal to dim(E(/z)); let (a_n)* be an orthonormal system in E such that, for the indices n for which X_n = /x_fc, the a_n constitute a basis of E(jn_fc). We say that (A_n) is the full sequence of strictly positive eigenvalues of u. (a) Show that A_n is the maximum value of (u(x) \ x) when x varies in the subset of E defined by the relations \|\|\|\| = 1, (x \ a_k)* = 0 for 1 < k ^ n — 1; furthermore, that maximum value is attained for x = a_n (use (11.5.7(d))). (b) Let z_i9..., z_n_i be arbitrary vectors in E, and denote by P_M(ZI, ..., ^«_i) the l.u.b. of (u(x)\x) when x varies in the subset of E defined by the relations HJC\|\| = 1, (\|z_fc) = 0for 1 ^k^n-1.* Show that X_n = p£ai₉...₉a_n-i)^p₁ti₁,..._tz_n-₁) (the " maximinimal principle "; take x in the subspace generated by a^,..., a_n and verifying the relations (x \ z_k) — 0 for 1 =£ k ^ n — 1). (c) Let «', u" be two compact self-adjoint operators, and suppose « = «'-f-«"; let (AO, (A£) be the full sequences of strictly positive eigenvalues of u' and u'\ respec- tively, (a'_n) and (a'n) the corresponding orthonormal systems. Show that if A_p, A£ and A_p+€_! are defined, then X_p+q^i < A£ 4- A£ (consider p_H(ai,..., «J_i, ai,..., a'q-i). If the sequence (Xn) is finite and has N terms, and if Ap and A_P+N are defined, then X_p+_N =s$ A; (same method, observing that (u"(x) \ x) ^ 0 if (x \ tf) = 0 for 1 ^ j ^ N). (d) Under the same assumptions as in (c)_s show that if A_p and X_p are defined, then \|A_P — A;^\| \|\|«\|\| (use the relation X_p = p_u(a_lt ...,a_p^i)).* Furthermore, if u"2Q* (resp. u'' ^ 0), then A_p ^ A_p (resp. A_p < A_p) (same method). (e) When E is finite dimensional, transcribe the results of (b), (c), (d) for hermitian forms on E x E (see Problem 3). Apply to the following problem: let /, (1 ^ i ^ n) be regulated functions in a compact interval I = [a, b], and let I' = [c, d] be an interval contained in I; let A = detQfifjdt}, A' = det/ f" f,fj dt\* be the Gram determinants corresponding to I and I⁷; show that A/ < A, by expressing the Gram determinants as products of eigenvalues. 9. (a) Let M be a compact self-adjoint operator in a complex Hilbert space E. Let H be a closed subspace of E, and p the orthogonal projection of E onto H (Section 6.3). Show that the restriction v to H of p o u (or ofp o u o _p) is compact and self-adjoint and that (v(y) \ y) = (u(y) \ y) for y e H (use the relation p* = p). Let (A_M), (^_n) be the full sequences of strictly positive eigenvalues of u and v respectively. Show that if A_nand fji_n are defined, then fi_M ^ X_n (use Problem 8(b)). (b) Suppose in addition u is positive. Show that for any finite sequence (x_k)^_k^_nof points of E, det((«(i) \| ,)) < AiA₂ • • • A_n det((x_t \ Xj)) (apply (a) to the subspace H generated by x_lt..., x_n). 10. (a) Let u be a hermitian operator in a complex prehilbert space E. Show that for any integer «>0, and any xeE, \\uⁿ(x)\\² < \\if~^l(x)\\ - \|\|M^W+I(JC)\|\| (use Cauchy- Schwarz).